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Relaxation effects in magnetic-field-induced martensitic transformation of an Ni–Mn–In–Co alloy
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ScienceDirect Acta Materialia 71 (2014) 117–125 www.elsevier.com/locate/actamat
Relaxation effects in magnetic-field-induced martensitic transformation of an Ni–Mn–In–Co alloy J.I. Pe´rez-Landaza´bal a, V. Recarte a, J. Torrens-Serra b, E. Cesari b,⇑ b
a Departamento de Fı´sica, Universidad Pu´blica de Navarra, Campus de Arrosadı´a, E-31006 Pamplona, Spain Departament de Fı´sica, Universitat de les Illes Balears, Cra. De Valldemossa km 7.5, E-07122 Palma de Mallorca, Spain
Received 25 October 2013; received in revised form 26 February 2014; accepted 28 February 2014
Abstract The effect of isothermal stepwise changes of an applied magnetic field in an Ni–Mn–In–Co metamagnetic shape memory alloy has been studied. It has been found that relaxations towards equilibrium states occur under isothermal and isofield conditions, in both direct and reverse martensitic transformations (upon decreasing or increasing field sequences, respectively). In all cases, the relaxation follows a logarithmic time dependence, which can be monitored through the time dependence of magnetization M(t). The logarithmic dependence brings close analogies with the thermal fluctuation magnetic after-effect, associated with the thermally assisted rearrangement of magnetic domains after a sudden variation of the magnetic field. Following this analogy, it has been found that the viscosity coefficient Z (slope of M(t) vs. ln t) is directly proportional to the transformation rate in a continuous cycle. This result generalizes what has been found in temperature-induced transformations, although in field-induced transformations the proportionality is temperature dependent. This result also allows for an understanding of the existence of isothermal–isofield “windows” where the relaxations can take place, and the experimental results found in quite different systems can be easily rationalized in this way. Finally, the features of pseudotransformation–temperature–time diagrams in this alloy can be rationalized according to the proposed dependence of Z. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Phase transformation kinetics; Metamagnetism; Martensitic phase transformation; Magnetic shape memory alloys
1. Introduction Ferromagnetic shape memory alloys (FSMAs) are a class of multiferroic materials which are currently generating a great deal of interest due to their unique properties, such as their ability to undergo giant magnetic-fieldinduced strains by variant reorientation, as in Ni–Mn–Ga [1], or associated with a magnetic-field-induced (reverse) transformation – a magnetic shape memory effect (MSM) – which take place in Ni–Mn–X type alloys (typically X = In, Sn, Sb) [2]. In these systems, a large inverse magnetocaloric effect has also been found [3,4]. Both this effect ⇑ Corresponding author.
E-mail address: [email protected] (E. Cesari).
and the MSM effect are related to the martensitic transformation (MT) accompanied by a large change in magnetic order between both phases. Even though it is commonly accepted that MTs have an athermal character, some cases of isothermal behaviour have been known for several years. For example, in some Fe–Ni–Mn systems, the application of high magnetic fields changes the originally isothermal kinetics to athermal [5]; in Cu–Al–Ni the MT occurs after some incubation period during isothermal holding at a temperature higher than the martensitic start temperature, TMs, and similarly, the reverse MT takes place upon isothermal holding at T < TAs [6]. Isothermal features are also present in the magnetostructural transition shown by Ni–Mn–X metamagnetic shape memory alloys [7–11]. Moreover, the effect of the
http://dx.doi.org/10.1016/j.actamat.2014.02.041 1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
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applied magnetic field seems to be very important in relation to the isothermal characteristics, showing some analogies with the above-mentioned Fe–Ni–Mn system, although it has to be kept in mind that the magnetic field favours the appearance of the martensitic phase in the Fe-based alloy, contrary to what occurs in Ni–Mn–X alloys. It has been recently reported that in Ni–Mn–X alloys the magnetic field leads to the arrest of the MT by stabilizing the parent (high magnetization) phase [7,8,12,13] and it is in the arrested MT where the isothermal behaviour is easily encountered. Nevertheless, Kustov et al. showed in an Ni–Mn–In–Co alloy that the arrested state is not a necessary condition for the isothermal behaviour [10]. In this sense, it is worth mentioning that very recently, isothermal MTs were found in Ni–Ti alloys, where no external field is applied; thus the isothermal behaviour does not have a necessary relation with any externally produced arrest of the transformation [14]. The progress of the isothermal transformation in Ni– Mn–X and Ni–Ti alloys which can be found by stopping the cooling/heating cycle within the respective transformation range, and/or by changing the magnetic field in the case of metamagnetic shape memory alloys, can be properly described by a logarithmic law, as has been shown in Refs. [10,14–17]. The logarithmic dependence brings close analogies with the thermal fluctuation magnetic after-effect, associated with the thermally assisted rearrangement of magnetic domains after a sudden variation of the magnetic field. It is considered that the overall metastable behaviour of the system, related to a complicated free energy surface landscape with a large number of local minima, can be decomposed into a superposition of many bistable contributions relaxing over a set of energy barriers [18,10]. In the case of relaxations in MT, the physical mechanism is different (as we are not dealing with pure rearrangements of magnetic domains), but the presence of different energy barriers through which the direct/reverse transformation takes place can be considered as an appropriate picture. From these previous experimental data it has been noticed that there is an interplay between the temperature and the magnetic field in the kinetics of the isothermal transformation through the slope of the logarithmic time dependence, Z [15,16]. In addition, some regularities have been observed in the relation between the martensite fraction present during the isothermal holding and the parameter Z, which could be ascribed to the role of martensite–austenite interfaces in the kinetics of the isothermal transformation [16]. Any possible application of shape memory alloys, whether ferromagnetic or not, often relies on the properties of an athermal MT; thus to clarify, the conditions which control the thermal/athermal kinetics are of obvious interest. Therefore, progress in the understanding of the isothermal/athermal character of the MTs in general, not only in the metamagnetic alloys, is necessary. The aim of this paper is to analyse in a more systematic way the isothermal transformation features produced by
isothermal (stepwise) changes of the applied magnetic field in an Ni–Mn–In–Co ferromagnetic shape memory alloy, at temperatures close or below the martensitic finish temperature, TMf, in such a way that sequences of increasing or decreasing applied magnetic field will promote the reverse or direct (isothermal) MT, respectively. These data are compared with the isothermal characteristics, obtained in the same alloy, by stopping cooling–heating temperatureinduced transformation cycles under a constant field. In the case of temperature-induced cycles, proportionality was found between the isothermal transformation rate, Z, and the transformation rate in a continuous cycle [10]. This relation is discussed and extended to isothermal, magnetic-field-induced MT. 2. Experimental procedure An Ni45.0Mn36.7In13.3Co5.0 (nominal composition, at.%) alloy produced by arc-melting, with several consecutive remeltings, has been used. Samples for magnetic measurements of parallelepiped shape of 0.5 1 4 mm3 were encapsulated in vacuum-sealed quartz tubes and further homogenized by 24 h annealing at 1170 K followed by water-quenching. After this treatment the alloy composition, as obtained by energy-dispersive X-ray spectroscopy, was Ni45.5Mn35.3In12.7Co6.5. It is known that the MT in these alloys can be shifted to lower temperatures by increasing the degree of L21 order [19] and that in this case the arrest of the transformation produced by the magnetic field is much more intense than in samples with a lower degree of order [10]. Therefore, the samples were subjected to an additional treatment of 15 min at 1070 K followed by slow cooling (hereafter referred to as SC samples – this naming has been chosen in order to be coherent with equivalent treatment in other papers). In this case TMs was close to 260 K and the structure of martensite was a mixture of seven-layered (14 M) and no-modulated (2 M) [9]. Absence of decomposition and/or precipitation was confirmed by optical and transmission electron microscopy. A Squid magnetometer [20] was used to measure the magnetization in series at constant temperature, under different applied fields, in the range 1–7 T. The time dependence of the magnetization M(t), which indicates the progress of the direct/ reverse transformation, is the physical magnitude from which the kinetics of the isothermal transformation is obtained. Once the temperature and the field are stable, M(t) follows a logarithmic law, in analogy with the results obtained by AC and DC resistance measurements [10]: MðtÞ ¼ M 0 þ Z lnð1 þ t=t0 Þ
ð1Þ
where M0 = M (t = 0) and t0, Z and M0 are the three parameters fitting the experimental data to the dependence given by Eq. (1). t0 takes into account possible transient effects and is related to the time required for the Squid to reach a constant value of the magnetic field after its variation. Following the analogy with the thermal fluctuation magnetic after-effect, Z can be considered equivalent to
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the magnetic viscosity. After zero field cooling down to the temperature of the experiment (120, 150 or 200 K), the magnetic field was changed in a stepwise mode. That is, after reaching the desired temperature a field of 1 T was applied and maintained isothermally a time interval long enough to reach a stationary M(t) value. After that, the field was decreased to 0.005 T. This procedure was repeated successively increasing in steps of 1 T each time, up to a maximum field value of 7 T (this is called increasing sequence). From 7 T the field was decreased in 1 T steps down to 1 T (decreasing sequence) following an analogous procedure. In this case, between two successive field-step values, a 7 T field is applied. In Fig. 2 an example of this experimental procedure measured at 200 K is presented, showing the different steps during the experiment. This procedure was carried out at 120, 150 and 200 K (which are below TMf in the absence of any field, but fall within the transformation range under the applied fields). In order to relate the isothermal kinetics with the relative advancement of the direct/reverse MT, it is necessary to follow the time evolution of transformed fraction of martensite given by: ðMðtÞ M min Þ ðM max MðtÞÞ X mart ðtÞ ¼ 1 ¼ ð2Þ ðM max M min Þ ðM max M min Þ where Mmax, the magnetization of the parent phase, depends on the applied field, namely at the start (TMs) or finish (TAf) temperatures of the MT. On the other hand, as Mmin is related to the magnetization of the martensite phase at TMf/TAs, a fixed value of Mmin corresponding to an applied field of 1 T has been taken, assuming that the arrest of the MT produced by this field is negligible. 3. Experimental results Fig. 1 shows the magnetization as a function of temperature during a cooling–heating cycle under different applied magnetic fields (0.01, 4 and 6 T). The curves reveal two characteristic phenomena of this type of alloy: the shift in
Fig. 1. Magnetization as a function of temperature for an Ni–Mn–In–Co SC sample under different applied magnetic fields: 0.001 T (black line), 4 T (blue line) and 6 T (red line).
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the transformation temperature range down to lower temperatures and the arrest of the martensitic transformation, by increasing the applied field from low field values (0.01 T) to high fields (4 and 6 T). Fig. 2 presents an example of the magnetization of the sample vs. time, M(t), due to the stepwise evolution of magnetic field which has been applied at different temperatures. When observing closely each field step, it is important to note the evolution of the magnetization at constant temperature and field from an initial magnetization value of M0, thus revealing a change of magnetization with time, a clear indication that the MT shows an isothermal behaviour. The M(t) curves for each field step have been fitted by a logarithmic dependence, as described in Eq. (1) and the rate constant Z is obtained for each case. Some examples are given in Fig. 3. In all cases, a good fitting to the dependence given by Eq. (1) has been obtained. Fig. 4a–c depicts the different Z values (emu g1) obtained for the different field-steps in both the decreasing and increasing magnetic field sequences at the three different studied temperatures. Fig. 4d–f shows the magnetization at the beginning of the constant field dwellings, M0, for each field step. Let us consider the increasing field case. At 200 K this sequence goes from a value of magnetization of 10 emu g1 corresponding to a sample with 100% martensite (see Fig. 4d, where the initial magnetization of the isofield steps is shown) up to a value of magnetization of 95 emu g1 at the beginning of the 7 T step corresponding to a martensite content of only 20% at this field. For this temperature case, a maximum value of Z 0.4 is observed at 6 T. For the experiments performed at 150 and 120 K, the increasing sequences start also at 100% martensite. It is important to keep in mind that there is no initial arrest of the transformation, as in this sequence the field is decreased to 0.005 T between successive isofield steps. However, at 150 K (see Fig. 4b) the Z values are higher than those at 200 K and increase continuously not showing any maximum up to 7 T but indicating a tendency to a maximum
Fig. 2. Time dependence of the magnetization under stepwise changes of the applied field at 200 K. Identical magnetic field vs. time sequences were applied at 120 and 150 K.
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(a)
(c)
(b)
(d)
Fig. 3. M(t) vs. time under field steps at 3 T and 6 T and 150 K; (a and c) within the decreasing field sequence, (b and d) within the increasing field sequence.
(a)
(d)
(b)
(e)
(c)
(f)
Fig. 4. (a–c) Z vs. field for the decreasing/increasing field sequences at 200, 150 and 120 K; The absolute values of Z are taken for the decreasing field case. (d–f) Magnetization at the beginning of the constant field dwellings. s Increasing field sequence, d decreasing field sequence.
value at higher fields. In this case, the sample evolves from fully martensitic state to about 30% austenite (which corresponds to a magnetization value of M0 40 emu g1) (see Fig. 4e). Similar trends can be observed at 120 K, but in this case the lower temperature just permits the transformation from 100% martensite to a less than 20% austenite (see Fig. 4c and f). In summary, the experimental results show that the reverse MT can take place isothermally under different fields, from initial states at 0.005 T that are practically coincident (as they are close to equilibrium states) at the three temperatures, reaching different amounts of austenite upon the stepwise increasing field change. Conversely, in the decreasing field sequence, the system goes from different initial states at 7 T (at the distinct temperatures, and through the intermediate field states), to a similar and close to the equilibrium state in the final step at 1 T. From the evolution under increasing fields up to 7 T, it is obvious that the maximum rate of isothermal reverse transformation is being shifted as the temperature decreases, thus the value of 7 T is not high enough to make visible the appearance of a maximum in the Z vs. field dependence at 150 and 120 K, whereas at 200 K is clearly seen at 6 T. On the other hand, a maximum in |Z| for the decreasing field sequence (black dotted curve in Fig. 4a–c) can be seen at the three temperatures considered. These differences are properly explained in the following discussion.
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4. Discussion Several aspects of the above results deserve a detailed discussion. First, let us consider the analogy with the magnetic after-effect, in which the magnetic viscosity Z is proportional to the product N{m}kBT: Z / N fmgk B T
ð3Þ
where N is the total number of relaxing units, {m} is the average change of magnetization per relaxing unit, kB is the Boltzmann constant and T the temperature. In a quite natural way, it can be considered that N is proportional to the amount of isothermally transforming material ({m} plays the role of an average transformation unit). Indeed, it has been found that the values of Z obtained at different temperatures (by dwellings within the transformation range) are proportional to the temperature rate of the formation of martensite (dF/dT) in an uninterrupted calorimetric cooling cycle, where F is the transformed fraction of martensite under these experimental conditions [10]. This means that dF/dT / N(T) / Z(T). In this case the T factor in the expression of Z has no significant effect, in agreement with the relatively narrow range of the temperature-induced transformation (40 K). On the other hand, in the results presented here it is obvious that the Z values are not scaling with temperature. Thus, let us examine the behaviour of Z/T / N{m}: considering first a global point of view in order to compare the distinct temperatures, it can be noticed that the mean values of Z, hZi, at the different temperatures have some correspondence with the average change in magnetization, hDMi, for both the increasing and decreasing sequences. However, even though it is clear the role of temperature in the values of Z and the related changes in magnetization, there is no strict proportionality between hZi/T and hDMi. At each temperature, a closer look to the different Z values makes evident the relation between Z or N{m} and the difference in magnetization between the previous stepwise field value and that for which Z is calculated. This difference can be properly approximated by the difference in the initial magnetization between two consecutive steps. It is easy to notice that the images in Fig. 4d–f have a close resemblance with the uninterrupted M(H) cycles at a given temperature. An example is given in Fig. 5, where a SC sample has been cycled, after zero field cooling, at 150 and 200 K. Therefore, from these experimental results it can be concluded that at a fixed temperature the rate of relaxation, Z, is proportional to the rate of change of magnetization in a continuous isothermal M(H) evolution: dF dM Z ¼ AðT Þ / ð4Þ dH T dH T The coefficient A(T) is explicitly included here to emphasize, as commented above, that the Z series do not scale with T, or, in other words, the proportionality coefficient relating Z and dF/dH|T (the field rate of the formation of martensite at constant T) is temperature dependent.
Fig. 5. Continuous M(H) cycles for the SC sample, after zero field cooling, at 150 and 200 K.
To understand this behaviour two aspects have to be considered. First, again in analogy with the magnetic aftereffect, when the field is abruptly changed, we may divide the “reaction” of the material in three groups [18]. One group contains the material (units) that remains in the same state after the field change (i.e. part of the sample that remains in austenite in the decreasing field sequence, because it continues to be in stable equilibrium under the new field value); a second group is formed by the volume elements of the material that change their state upon the field change. These are the units that “instantaneously” transform (i.e. athermally transforming units from austenite to martensite when the field is step-like reduced). Finally, the third group consists of the transforming units for which a stable initial state becomes metastable when the field is changed. These are the units that contribute to the relaxation and transform isothermally and at constant field. The second aspect is to take into account the experimental procedure of field step-like variations: for both the increasing/decreasing series a low (0.005 T)/high (7 T) field has been applied between two successive step field values. The time elapsed between the intermediate low/high field value and the next constant step for the field is long enough to allow for the global change in magnetization that can take place due to the step field change (mostly in “instantaneous” or athermal mode); according to the observed magnetization changes, the isothermally transformed fraction at constant field is much lower than the athermal one during the step field change (see Fig. 2). Therefore, the relaxation rate or the number of relaxing units (Z / N{m}) appears to be proportional to the difference between the “new” transformed units upon the application of the field and those that have transformed under the previous stepwise field value. It is worth noting that Z is not proportional to the total change in magnetization from the high/low field to the selected constant field value. This difference essentially corresponds to the field rate of the transformation, dF/dH|T, in a continuous field increasing/
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decreasing constant temperature process inducing reverse/ direct MT. As commented above, Fig. 5 shows examples of continuous M(H) cycles at constant temperature. From these data, the derivative dM/dH can be obtained numerically. Fig. 6 presents the comparison between Z series at the different temperatures and the values of dM/dH. Therefore, the relation between Z and dF/dH|T is confirmed, as established in Eq. (4). In this sense, within this more “detailed” point of view, there is a clear parallelism in the relation between Z and the transformation rate in a continuous evolution of the field at constant temperature (Z(H) / dF/dH|T), and the previously found proportionality between Z (from relaxation in dwellings at constant temperature, constant field) and the transformation rate dF/dT|H in a continuous cooling/heating cycle. Therefore the relation between Z(T) and the rate dF/dT|H in temperature-induced transformations is found to be extended to a similar relation between Z(H) and dF/dH|T. This similarity anticipates a universal physical background in both temperature-driven and field-driven anisothermal transformations, and sets the close relationship between the isothermal transformation rate (Z / N{m}) when it is observable, and the total transformation rate dF/dT|H or dF/dH|T in the anisothermal transformations according to the definition given by Massalski [21]. Another point that deserves further discussion is linked to the presence of an extreme in the isothermal transformation rates (or a maximum considering the absolute values of Z in the decreasing field sequence). As has been mentioned above, the maximum is clearly seen for the three temperatures in the decreasing field sequence; nevertheless, it can be assumed that such a maximum also exists in the increasing field case: it is found at 200 K (see Fig. 4a), and the evolution of Z with the field up to 7 T at 150
(a)
and 120 K, anticipates that a maximum will be present at higher fields (in agreement with what has been discussed in the previous paragraph, dF/dH|T has to be bounded, as is the magnetization). These dependences of Z with the increasing/decreasing field sequences are perfectly coherent with the relation found between Z(H) and the ratio dF/ dH|T. The presence of a maximum in the values of Z means that there are limits for the field values, bounded at two sides, out of which the isothermal or relaxation behaviour does not take place, or is not observable at the level of the time scale allowed for the experimental conditions to reach a stepwise level of the field. In the present case, the isothermal isofield relaxation intervals (isothermal “windows”) for increasing/decreasing fields appear to have some degree of superposition in the range of studied temperatures. It is interesting to notice that the existence of isothermal windows (with logarithmic relaxation rates) has been found in a quite different system as Ce(Fe0.96Ru0.04)2 which also shows a metamagnetic transition (the low temperature phase has lower magnetization), as in the present alloy [22]. According to the shape of the M(H) curves of this alloy – taken at 5 K – in a continuous field change (see Figs. 3 and 6 of Ref. [22]), the isothermal behaviour window in increasing field sequences is found at higher fields and with a sharper cut-off limit than the corresponding window for the decreasing field sequence. These features are perfectly described taking into account the relation between Z(H) and dF/dH (Eq. (4)), which has been found in the present work for the Ni–Mn–In–Co alloy. Moreover, it is not surprising that this Ce-containing alloy presents disjoint relaxation windows for increasing/decreasing fields because of the much higher values of dM/dH (30 emu g1 T1), as compared to the present Ni–Mn– In–Co alloy, in most part of the M(H) curves. Taking into
(b)
(c)
(d)
Fig. 6. Comparison between Z and dM/dH obtained from continuous M(H) cycles as those shown in Fig. 5: (a) 200 K, H decreasing; (b) 200 K, H increasing; (c) 150 K, H decreasing; and (d) 150 K, H increasing.
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account the hysteresis in the M(H) cycle, this leads to a separate intervals of relaxation, again in agreement with Z(H) / dF/dH (Eq. (4)). It is necessary to mention here that for the same Ni– Mn–In–Co alloy, in temperature-induced transformations under constant field, relaxation dependences have been found, either in both or just in the cooling branch [9,10,16]. Therefore, it can be stated that the existence of certain (DT, DH) intervals of the driving variables for which the isothermal transformation occurs, is a general feature. Given a definite state of the system (including structure, degree of order, defects configuration, etc., which define the barriers to be overcome upon transformation, or in the thermodynamic sense, once the DG(T, H) landscape is determined), the relaxation can be observed or not, depending on the time dependence imposed to the variables T and H, and also on the available experimental time resolution for T and H. A very simple and typical example is given by the anisothermal transformations, which can be seen as athermal in continuous cooling/heating runs (for certain intervals of dT/dt under constant field, or dH/dt at constant T) or showing isothermal transformation steps at certain temperature (or field) dwellings within the continuous transformation range. It has to be taken into account that the transformation path F(T) or F(H) and consequently dF/dT or dF/dH can be modified depending on the dT/dt (dH/dt) rates, as the transformation includes time-dependent relaxations taking place during the continuous transformation (T induced or H induced). In the present Ni–Mn–In–Co alloy, under dT/ dt in the order of 10 K min1, or dH/dt 0.01 T s1 no significant changes in the transformation path are observed; the same occurs in the Ce-containing alloy [22]. This behaviour will be observed as far as the “athermal” component is contributing much more than the isothermal or relaxation one to the transformed fraction. Very recently, metastable states which relax towards the equilibrium have been found in an Ni–Co–Mn–Sn alloy [23]. According to this paper, these metastable states at low temperature are governed by the interplay of supercooled and kinetically arrested states, because none of them is the equilibrium state of the system. Therefore, is not surprising that under these conditions the system presents isothermal transformation as a mechanism to approach the equilibrium. The authors define the so-called kinetic arrest “bands”, given by some pairs of (H, T) values. The relations between these arrest bands and the isothermal or relaxation windows will be the object of future research. The fact that the number of relaxing units appears to be proportional to the global transformation rate, either dF/ dH|T or dF/dT|H, means that, upon a change in the driving field (T or H), there is no “a priori” separation between the parts of the sample (“units”) behaving in athermal (“instantaneous”) mode and those relaxing. Indeed, as they occur in the anisothermal transformations, relaxing units may become athermal or vice versa, by simply modifying
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the rate of variation in the driving fields, thus changing in a different way the free energy balance of the system. This feature is in perfect agreement with the common underlying physical mechanism of the phase transition in both types of units/transformation modes. At a macroscopic level, this means, for example, that the continuous transformation path is encountered once the isothermal or isofield dwellings are resumed, as has been observed in temperature-induced transformations in the same Ni– Mn–In–Co alloy considered here [10]. The relation between Z and the transformation rate allows for analysing some characteristics of the relaxation as a function of the progress of the transformation, i.e. as a function of the martensite fraction Xmart. This fraction, as mentioned above, can be easily calculated, for example at the beginning of the isofield dwellings, through the magnetization of the sample. In temperature-induced transformations under constant field, the construction of the socalled pseudo-transformation–temperature–time (TTT) diagrams, plotting martensite fraction as a function of 1/ |Z| has proved to be useful [16]. These diagrams are inspired in the conventional TTT diagrams used to characterize isothermal transformations, where every point (temperature–time pair) represents the condition to start the transformation and to obtain a given amount – usually a few per cent – of transformed material. In the pseudoTTT diagrams, the abscissa or time axis is replaced by the values of 1/|Z| as representative of the time rate at which the relaxation takes place. It has to be noticed that the pairs (temperature, 1/|Z|) represent the rate of accumulation of isothermally, isofield transformed phase for rather long time periods and at different degrees of the MT progress. In conventional TTT plots only the beginning of the MT and a small transformed fraction are considered. Pseudo-TTT diagrams have been obtained for the same Ni–Mn–In–Co alloy, for temperature-induced MT under constant fields [16]. It has been found that the pseudoTTT plot representing the martensite fraction as a function of 1/|Z| appears to be a common C-shaped curve, ranging from (practically) 0% to 100% transformed fraction, which includes different temperatures (Z values obtained at different martensite fractions in both direct and reverse MT), and applied magnetic fields. This case corresponds to a sample (water quenched, WQ, with a lower order degree than the SC samples used in the present work) that does not show arrest of the MT, at least up to the highest applied fields (4 T) in Ref. [16]. The nose of the common C-curve is situated very close to 50% of martensite fraction. On the other hand, in a continuous cycle, including complete direct/reverse MT, the maximum transformation rate corresponds to the maximum slope dM(T)/dT (temperature-induced MT at constant field, monitored by M(T)) or dR/dT (MT monitored by electrical resistance (R) vs. temperature measurements (T)); as this zone is close to 50% transformation, we found again the relation between Z and dF/dT, now in particular for |Z|max and dF/dT|max. This has been related to the amount of “active”
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austenite–martensite interfaces, which is largest close to 50% transformation in a complete cycle [16]. Fig. 7a compares the martensite fraction vs. 1/|Z| for a temperatureinduced MT under constant field (1 T) in a SC sample, with the SC sample subjected to the increasing/decreasing field procedure at 200 K. It is worth noting the similarity between these curves, even though the direct MT, in the decreasing field sequence, does not go from 0% to 100% martensite, as it starts at 80% austenite. It is obvious that the nose is close to 50% martensite. Looking at the increasing field points in Fig. 7a, they follow the same trend, taking into account that a field of 7 T is not enough to complete the reverse MT. However, the C-shaped curve for these points is shifted to higher values of 1/Z as compared with the decreasing field one, in correspondence with the values of Fig. 4a. From the results presented in Fig. 7a it could be concluded that for transformation paths, either T-induced or H-induced, which comprise a significant transformation amount (i.e. change in the martensite fraction of 0.8 or higher), the nose or maximum of |Z|
(a)
(b)
Fig. 7. Martensite fraction (Xmart) vs. 1/|Z| for temperature-induced transformation under constant field (1 T), and for increasing/decreasing fields at (a) 200 K and (b) 150 K.
is close to a 50% martensite. As has been stated in Ref. [16] for T-induced MT under constant field, this means that the maximum number of transforming units is related to the largest area/number of moving martensite–austenite interfaces, which corresponds to 50% transformation in a complete (or close to complete) transformation cycle. This is one of the significant points of the present work, as these results generalize to H-induced transformations those of Ref. [16] obtained for temperature-induced MT at constant fields. This fact means that a common physical mechanism is responsible for the time-dependent transformations. Let us discuss the relation between the nose position (|Z|max) in pseudo-TTT diagrams and the fractions of martensite in the cases were the amount of transformation is reduced. Fig. 7b shows again Xmart vs. 1/|Z| for the increasing and decreasing field sequences at 150 K, together with the data of MT under constant field (1 T) in a SC sample, already shown in Fig. 7a, for comparison. First, it can be noticed that the increasing field series at 150 K follows the same C-shaped curve as for 200 K, although the minimum of 1/|Z| does not correspond to the nose. This is in agreement with the fact that a field of 7 T is far from being able to complete the reverse transformation at 150 K, and the maximum of Z is not reached at this temperature (as seen in Fig. 4b). The same trends are observed at 120 K. For the decreasing field sequence a kind of nose appears. Even though the total fraction of transformed martensite is not far from 0.3, a maximum of |Z| appears in correspondence with the shape of the decreasing field curve, as seen in Fig. 5 (the shape of the increasing field curve obviously explains that up to 7 T a maximum slope is not reached, and therefore the absence of a nose), in agreement with Eq. (4). Keeping once more in mind the relation between Z and N, it can be concluded that, even in the cases where the total transformed fraction is low, the value of Z (/ N / dF/dH) is related to the number of “active” martensite–austenite interfaces. 5. Conclusions Relaxation towards the equilibrium state occurs upon isothermal magnetic field step variations, both in increasing and decreasing field sequences. In all cases the time dependence, followed through the magnetization M(t), appears to be logarithmic. Therefore, the analogy with the magnetic after-effect allows analysing the slope of logarithmic dependence, Z (or viscosity coefficient), in terms of the number of relaxing units and temperature (Eq. (3)). It has been found that the rate of relaxation or the number of relaxing units (Z / N{m}) appears to be proportional to the transformation rate upon isothermal, continuous field change Z(H) / dF/dH|T / dM/dH|T. Therefore the previously established proportionality between Z(T) (relaxation in dwellings at constant temperature within a temperature cycle), and the transformation rate dF/dT|H in a continuous cooling/heating cycle, is
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found to be generalized to a similar relation between Z(H) and dF/dH|T. This fact points out a common physical origin of relaxations in both temperature- and field-induced MT. Following the analogy with the magnetic after-effect, the relaxations take place through a variety of energy barriers, leading to logarithmic time dependence. The presence of a maximum in Z sequences implies that there are limits for the field values, bounded at two sides, out of which the isothermal/isofield or relaxation behaviour does not take place, or is not observable at the level of the time scale allowed for the experimental requirements to reach a stepwise level of the field. In the present case, the isothermal isofield relaxation intervals (isothermal “windows”) for increasing/decreasing fields appear to have some degree of superposition in the range of the studied temperatures. The existence of isothermal windows has been found in different systems as Ce(Fe0.96Ru0.04)2. The relaxation features of this system are perfectly explained taking into account the relation between Z(H) and dF/ dH, which has been found for the present Ni–Mn–In–Co alloy. The relation between Z and the transformation rate allows for analysing some characteristics of the relaxation as a function of the progress of the transformation, i.e. as a function of the martensite fraction. As in temperature-induced transformations under constant field, the socalled pseudo-TTT diagrams can be constructed in the case of isothermal/isofield relaxations. The relation between the maximum relaxation rate (minimum of 1/|Z| or nose in the TTT diagram) and the number of interfaces appears to be also valid for the present case, thus extending and generalizing to magnetic-field-induced MT the results found in temperature-induced transformations under constant field. In summary, relaxations towards the equilibrium have been found to have a common physical background in both temperature-driven (Z(T) / dF/dT|H) and field-driven (Z(H) / dF/dH|T) transformations. The existence of relaxation windows has been understood directly from the experimental results, and the shape of pseudo-TTT diagrams has been related to the number of “active” units participating in the relaxation. Acknowledgements Partial financial support from MINECO and FEDER (Projects MAT2011-28217-C02-01 and MAT2012-37923-
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C02-01) is acknowledged. Valuable comments of Prof. Victor A. L’vov are also acknowledged. References [1] Ullakko K, Huang JK, Kantner C, O’Handley RC, Kokorin VV. Appl Phys Lett 1996;69:1966. [2] Kainuma R, Imano Y, Ito W, Sutou Y, Morito H, Okamoto S, et al. Nature 2006;439:957. [3] Sharma VK, Chattopadhyay MK, Roy SB. J Phys D Appl Phys 2007;40:1869. [4] Liu J, Gottschall T, Skokov KP, Moore JD, Gutfleisch O. Nat Mater 2012;11:620. [5] Kakeshita T, Kuriowa K, Shimizu K, Ikeda T, Yamagishi A, Date M. Mater Trans JIM 1993;34:415. [6] Kakeshita T, Takeguchi T, Fukuda T, Saburi T. Mater Trans JIM 1996;37:299. [7] Sharma K, Chattopadhyay MK, Roy SB. Phys Rev B 2007;76:140401. [8] Chatterjee S, Giri S, Majumdar S, De SK. Phys Rev B 2008;77:224440. [9] Kustov S, Corro´ ML, Pons J, Cesari E. Appl Phys Lett 2009;94:191901. [10] Kustov S, Golovin I, Corro´ ML, Cesari E. J Appl Phys 2010;107:053525. [11] Lee Y, Todai M, Okuyama T, Fukuda T, Kakeshita T, Kainuma R. Scripta Mater 2011;64:927. [12] Ito W, Ito K, Umetsu RY, Kainuma R, Koyama K, Watanabe K, et al. Appl Phys Lett 2008;92:021908. [13] Umetsu RY, Ito W, Ito K, Koyama K, Fujita A, Oikawa K, et al. Scripta Mater 2009;60:25. [14] Kustov S, Salas D, Cesari E, Santamarta R, Van Humbeeck J. Acta Mater 2012;60:2578. [15] Pe´rez-Landaza´bal JI, Recarte V, Sa´nchez-Alarcos V, Go´mez Polo C, Kustov S, Cesari E. J Appl Phys 2011;109:093515. [16] Pe´rez-Landaza´bal JI, Recarte V, Sa´nchez-Alarcos V, Kustov S, Salas D, Cesari E. Intermetallics 2012;28:144. [17] Pe´rez-Landaza´bal JI, Recarte V, Sa´nchez-Alarcos V, Kustov S, Cesari E. J Alloys Compd 2012;536S:S277. [18] Bertotti G. Hysteresis in magnetism. San Diego, CA: Academic Press; 1998. [19] Recarte V, Pe´rez-Landaza´bal JI, Sa´nchez-Alarcos V, Rodrı´guezVelamaza´n JA. Acta Mater 2012;60:1937. [20] UNPN-E0007 Feder Project; 2003. [21] Massalski TB. Mater Trans 2010;51:583. [22] Chattopadhyay MK, Roy SB, Nigam AK, Sokhey KJS, Chadda P. Phys Rev B 2003;68:174404. [23] Lakhani A, Banerjee A, Chaddah P, Chen X, Ramanujan RV. J Phys Condens Matter 2012;24:386004.
ScienceDirect Acta Materialia 71 (2014) 117–125 www.elsevier.com/locate/actamat
Relaxation effects in magnetic-field-induced martensitic transformation of an Ni–Mn–In–Co alloy J.I. Pe´rez-Landaza´bal a, V. Recarte a, J. Torrens-Serra b, E. Cesari b,⇑ b
a Departamento de Fı´sica, Universidad Pu´blica de Navarra, Campus de Arrosadı´a, E-31006 Pamplona, Spain Departament de Fı´sica, Universitat de les Illes Balears, Cra. De Valldemossa km 7.5, E-07122 Palma de Mallorca, Spain
Received 25 October 2013; received in revised form 26 February 2014; accepted 28 February 2014
Abstract The effect of isothermal stepwise changes of an applied magnetic field in an Ni–Mn–In–Co metamagnetic shape memory alloy has been studied. It has been found that relaxations towards equilibrium states occur under isothermal and isofield conditions, in both direct and reverse martensitic transformations (upon decreasing or increasing field sequences, respectively). In all cases, the relaxation follows a logarithmic time dependence, which can be monitored through the time dependence of magnetization M(t). The logarithmic dependence brings close analogies with the thermal fluctuation magnetic after-effect, associated with the thermally assisted rearrangement of magnetic domains after a sudden variation of the magnetic field. Following this analogy, it has been found that the viscosity coefficient Z (slope of M(t) vs. ln t) is directly proportional to the transformation rate in a continuous cycle. This result generalizes what has been found in temperature-induced transformations, although in field-induced transformations the proportionality is temperature dependent. This result also allows for an understanding of the existence of isothermal–isofield “windows” where the relaxations can take place, and the experimental results found in quite different systems can be easily rationalized in this way. Finally, the features of pseudotransformation–temperature–time diagrams in this alloy can be rationalized according to the proposed dependence of Z. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Phase transformation kinetics; Metamagnetism; Martensitic phase transformation; Magnetic shape memory alloys
1. Introduction Ferromagnetic shape memory alloys (FSMAs) are a class of multiferroic materials which are currently generating a great deal of interest due to their unique properties, such as their ability to undergo giant magnetic-fieldinduced strains by variant reorientation, as in Ni–Mn–Ga [1], or associated with a magnetic-field-induced (reverse) transformation – a magnetic shape memory effect (MSM) – which take place in Ni–Mn–X type alloys (typically X = In, Sn, Sb) [2]. In these systems, a large inverse magnetocaloric effect has also been found [3,4]. Both this effect ⇑ Corresponding author.
E-mail address: [email protected] (E. Cesari).
and the MSM effect are related to the martensitic transformation (MT) accompanied by a large change in magnetic order between both phases. Even though it is commonly accepted that MTs have an athermal character, some cases of isothermal behaviour have been known for several years. For example, in some Fe–Ni–Mn systems, the application of high magnetic fields changes the originally isothermal kinetics to athermal [5]; in Cu–Al–Ni the MT occurs after some incubation period during isothermal holding at a temperature higher than the martensitic start temperature, TMs, and similarly, the reverse MT takes place upon isothermal holding at T < TAs [6]. Isothermal features are also present in the magnetostructural transition shown by Ni–Mn–X metamagnetic shape memory alloys [7–11]. Moreover, the effect of the
http://dx.doi.org/10.1016/j.actamat.2014.02.041 1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
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applied magnetic field seems to be very important in relation to the isothermal characteristics, showing some analogies with the above-mentioned Fe–Ni–Mn system, although it has to be kept in mind that the magnetic field favours the appearance of the martensitic phase in the Fe-based alloy, contrary to what occurs in Ni–Mn–X alloys. It has been recently reported that in Ni–Mn–X alloys the magnetic field leads to the arrest of the MT by stabilizing the parent (high magnetization) phase [7,8,12,13] and it is in the arrested MT where the isothermal behaviour is easily encountered. Nevertheless, Kustov et al. showed in an Ni–Mn–In–Co alloy that the arrested state is not a necessary condition for the isothermal behaviour [10]. In this sense, it is worth mentioning that very recently, isothermal MTs were found in Ni–Ti alloys, where no external field is applied; thus the isothermal behaviour does not have a necessary relation with any externally produced arrest of the transformation [14]. The progress of the isothermal transformation in Ni– Mn–X and Ni–Ti alloys which can be found by stopping the cooling/heating cycle within the respective transformation range, and/or by changing the magnetic field in the case of metamagnetic shape memory alloys, can be properly described by a logarithmic law, as has been shown in Refs. [10,14–17]. The logarithmic dependence brings close analogies with the thermal fluctuation magnetic after-effect, associated with the thermally assisted rearrangement of magnetic domains after a sudden variation of the magnetic field. It is considered that the overall metastable behaviour of the system, related to a complicated free energy surface landscape with a large number of local minima, can be decomposed into a superposition of many bistable contributions relaxing over a set of energy barriers [18,10]. In the case of relaxations in MT, the physical mechanism is different (as we are not dealing with pure rearrangements of magnetic domains), but the presence of different energy barriers through which the direct/reverse transformation takes place can be considered as an appropriate picture. From these previous experimental data it has been noticed that there is an interplay between the temperature and the magnetic field in the kinetics of the isothermal transformation through the slope of the logarithmic time dependence, Z [15,16]. In addition, some regularities have been observed in the relation between the martensite fraction present during the isothermal holding and the parameter Z, which could be ascribed to the role of martensite–austenite interfaces in the kinetics of the isothermal transformation [16]. Any possible application of shape memory alloys, whether ferromagnetic or not, often relies on the properties of an athermal MT; thus to clarify, the conditions which control the thermal/athermal kinetics are of obvious interest. Therefore, progress in the understanding of the isothermal/athermal character of the MTs in general, not only in the metamagnetic alloys, is necessary. The aim of this paper is to analyse in a more systematic way the isothermal transformation features produced by
isothermal (stepwise) changes of the applied magnetic field in an Ni–Mn–In–Co ferromagnetic shape memory alloy, at temperatures close or below the martensitic finish temperature, TMf, in such a way that sequences of increasing or decreasing applied magnetic field will promote the reverse or direct (isothermal) MT, respectively. These data are compared with the isothermal characteristics, obtained in the same alloy, by stopping cooling–heating temperatureinduced transformation cycles under a constant field. In the case of temperature-induced cycles, proportionality was found between the isothermal transformation rate, Z, and the transformation rate in a continuous cycle [10]. This relation is discussed and extended to isothermal, magnetic-field-induced MT. 2. Experimental procedure An Ni45.0Mn36.7In13.3Co5.0 (nominal composition, at.%) alloy produced by arc-melting, with several consecutive remeltings, has been used. Samples for magnetic measurements of parallelepiped shape of 0.5 1 4 mm3 were encapsulated in vacuum-sealed quartz tubes and further homogenized by 24 h annealing at 1170 K followed by water-quenching. After this treatment the alloy composition, as obtained by energy-dispersive X-ray spectroscopy, was Ni45.5Mn35.3In12.7Co6.5. It is known that the MT in these alloys can be shifted to lower temperatures by increasing the degree of L21 order [19] and that in this case the arrest of the transformation produced by the magnetic field is much more intense than in samples with a lower degree of order [10]. Therefore, the samples were subjected to an additional treatment of 15 min at 1070 K followed by slow cooling (hereafter referred to as SC samples – this naming has been chosen in order to be coherent with equivalent treatment in other papers). In this case TMs was close to 260 K and the structure of martensite was a mixture of seven-layered (14 M) and no-modulated (2 M) [9]. Absence of decomposition and/or precipitation was confirmed by optical and transmission electron microscopy. A Squid magnetometer [20] was used to measure the magnetization in series at constant temperature, under different applied fields, in the range 1–7 T. The time dependence of the magnetization M(t), which indicates the progress of the direct/ reverse transformation, is the physical magnitude from which the kinetics of the isothermal transformation is obtained. Once the temperature and the field are stable, M(t) follows a logarithmic law, in analogy with the results obtained by AC and DC resistance measurements [10]: MðtÞ ¼ M 0 þ Z lnð1 þ t=t0 Þ
ð1Þ
where M0 = M (t = 0) and t0, Z and M0 are the three parameters fitting the experimental data to the dependence given by Eq. (1). t0 takes into account possible transient effects and is related to the time required for the Squid to reach a constant value of the magnetic field after its variation. Following the analogy with the thermal fluctuation magnetic after-effect, Z can be considered equivalent to
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the magnetic viscosity. After zero field cooling down to the temperature of the experiment (120, 150 or 200 K), the magnetic field was changed in a stepwise mode. That is, after reaching the desired temperature a field of 1 T was applied and maintained isothermally a time interval long enough to reach a stationary M(t) value. After that, the field was decreased to 0.005 T. This procedure was repeated successively increasing in steps of 1 T each time, up to a maximum field value of 7 T (this is called increasing sequence). From 7 T the field was decreased in 1 T steps down to 1 T (decreasing sequence) following an analogous procedure. In this case, between two successive field-step values, a 7 T field is applied. In Fig. 2 an example of this experimental procedure measured at 200 K is presented, showing the different steps during the experiment. This procedure was carried out at 120, 150 and 200 K (which are below TMf in the absence of any field, but fall within the transformation range under the applied fields). In order to relate the isothermal kinetics with the relative advancement of the direct/reverse MT, it is necessary to follow the time evolution of transformed fraction of martensite given by: ðMðtÞ M min Þ ðM max MðtÞÞ X mart ðtÞ ¼ 1 ¼ ð2Þ ðM max M min Þ ðM max M min Þ where Mmax, the magnetization of the parent phase, depends on the applied field, namely at the start (TMs) or finish (TAf) temperatures of the MT. On the other hand, as Mmin is related to the magnetization of the martensite phase at TMf/TAs, a fixed value of Mmin corresponding to an applied field of 1 T has been taken, assuming that the arrest of the MT produced by this field is negligible. 3. Experimental results Fig. 1 shows the magnetization as a function of temperature during a cooling–heating cycle under different applied magnetic fields (0.01, 4 and 6 T). The curves reveal two characteristic phenomena of this type of alloy: the shift in
Fig. 1. Magnetization as a function of temperature for an Ni–Mn–In–Co SC sample under different applied magnetic fields: 0.001 T (black line), 4 T (blue line) and 6 T (red line).
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the transformation temperature range down to lower temperatures and the arrest of the martensitic transformation, by increasing the applied field from low field values (0.01 T) to high fields (4 and 6 T). Fig. 2 presents an example of the magnetization of the sample vs. time, M(t), due to the stepwise evolution of magnetic field which has been applied at different temperatures. When observing closely each field step, it is important to note the evolution of the magnetization at constant temperature and field from an initial magnetization value of M0, thus revealing a change of magnetization with time, a clear indication that the MT shows an isothermal behaviour. The M(t) curves for each field step have been fitted by a logarithmic dependence, as described in Eq. (1) and the rate constant Z is obtained for each case. Some examples are given in Fig. 3. In all cases, a good fitting to the dependence given by Eq. (1) has been obtained. Fig. 4a–c depicts the different Z values (emu g1) obtained for the different field-steps in both the decreasing and increasing magnetic field sequences at the three different studied temperatures. Fig. 4d–f shows the magnetization at the beginning of the constant field dwellings, M0, for each field step. Let us consider the increasing field case. At 200 K this sequence goes from a value of magnetization of 10 emu g1 corresponding to a sample with 100% martensite (see Fig. 4d, where the initial magnetization of the isofield steps is shown) up to a value of magnetization of 95 emu g1 at the beginning of the 7 T step corresponding to a martensite content of only 20% at this field. For this temperature case, a maximum value of Z 0.4 is observed at 6 T. For the experiments performed at 150 and 120 K, the increasing sequences start also at 100% martensite. It is important to keep in mind that there is no initial arrest of the transformation, as in this sequence the field is decreased to 0.005 T between successive isofield steps. However, at 150 K (see Fig. 4b) the Z values are higher than those at 200 K and increase continuously not showing any maximum up to 7 T but indicating a tendency to a maximum
Fig. 2. Time dependence of the magnetization under stepwise changes of the applied field at 200 K. Identical magnetic field vs. time sequences were applied at 120 and 150 K.
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(a)
(c)
(b)
(d)
Fig. 3. M(t) vs. time under field steps at 3 T and 6 T and 150 K; (a and c) within the decreasing field sequence, (b and d) within the increasing field sequence.
(a)
(d)
(b)
(e)
(c)
(f)
Fig. 4. (a–c) Z vs. field for the decreasing/increasing field sequences at 200, 150 and 120 K; The absolute values of Z are taken for the decreasing field case. (d–f) Magnetization at the beginning of the constant field dwellings. s Increasing field sequence, d decreasing field sequence.
value at higher fields. In this case, the sample evolves from fully martensitic state to about 30% austenite (which corresponds to a magnetization value of M0 40 emu g1) (see Fig. 4e). Similar trends can be observed at 120 K, but in this case the lower temperature just permits the transformation from 100% martensite to a less than 20% austenite (see Fig. 4c and f). In summary, the experimental results show that the reverse MT can take place isothermally under different fields, from initial states at 0.005 T that are practically coincident (as they are close to equilibrium states) at the three temperatures, reaching different amounts of austenite upon the stepwise increasing field change. Conversely, in the decreasing field sequence, the system goes from different initial states at 7 T (at the distinct temperatures, and through the intermediate field states), to a similar and close to the equilibrium state in the final step at 1 T. From the evolution under increasing fields up to 7 T, it is obvious that the maximum rate of isothermal reverse transformation is being shifted as the temperature decreases, thus the value of 7 T is not high enough to make visible the appearance of a maximum in the Z vs. field dependence at 150 and 120 K, whereas at 200 K is clearly seen at 6 T. On the other hand, a maximum in |Z| for the decreasing field sequence (black dotted curve in Fig. 4a–c) can be seen at the three temperatures considered. These differences are properly explained in the following discussion.
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4. Discussion Several aspects of the above results deserve a detailed discussion. First, let us consider the analogy with the magnetic after-effect, in which the magnetic viscosity Z is proportional to the product N{m}kBT: Z / N fmgk B T
ð3Þ
where N is the total number of relaxing units, {m} is the average change of magnetization per relaxing unit, kB is the Boltzmann constant and T the temperature. In a quite natural way, it can be considered that N is proportional to the amount of isothermally transforming material ({m} plays the role of an average transformation unit). Indeed, it has been found that the values of Z obtained at different temperatures (by dwellings within the transformation range) are proportional to the temperature rate of the formation of martensite (dF/dT) in an uninterrupted calorimetric cooling cycle, where F is the transformed fraction of martensite under these experimental conditions [10]. This means that dF/dT / N(T) / Z(T). In this case the T factor in the expression of Z has no significant effect, in agreement with the relatively narrow range of the temperature-induced transformation (40 K). On the other hand, in the results presented here it is obvious that the Z values are not scaling with temperature. Thus, let us examine the behaviour of Z/T / N{m}: considering first a global point of view in order to compare the distinct temperatures, it can be noticed that the mean values of Z, hZi, at the different temperatures have some correspondence with the average change in magnetization, hDMi, for both the increasing and decreasing sequences. However, even though it is clear the role of temperature in the values of Z and the related changes in magnetization, there is no strict proportionality between hZi/T and hDMi. At each temperature, a closer look to the different Z values makes evident the relation between Z or N{m} and the difference in magnetization between the previous stepwise field value and that for which Z is calculated. This difference can be properly approximated by the difference in the initial magnetization between two consecutive steps. It is easy to notice that the images in Fig. 4d–f have a close resemblance with the uninterrupted M(H) cycles at a given temperature. An example is given in Fig. 5, where a SC sample has been cycled, after zero field cooling, at 150 and 200 K. Therefore, from these experimental results it can be concluded that at a fixed temperature the rate of relaxation, Z, is proportional to the rate of change of magnetization in a continuous isothermal M(H) evolution: dF dM Z ¼ AðT Þ / ð4Þ dH T dH T The coefficient A(T) is explicitly included here to emphasize, as commented above, that the Z series do not scale with T, or, in other words, the proportionality coefficient relating Z and dF/dH|T (the field rate of the formation of martensite at constant T) is temperature dependent.
Fig. 5. Continuous M(H) cycles for the SC sample, after zero field cooling, at 150 and 200 K.
To understand this behaviour two aspects have to be considered. First, again in analogy with the magnetic aftereffect, when the field is abruptly changed, we may divide the “reaction” of the material in three groups [18]. One group contains the material (units) that remains in the same state after the field change (i.e. part of the sample that remains in austenite in the decreasing field sequence, because it continues to be in stable equilibrium under the new field value); a second group is formed by the volume elements of the material that change their state upon the field change. These are the units that “instantaneously” transform (i.e. athermally transforming units from austenite to martensite when the field is step-like reduced). Finally, the third group consists of the transforming units for which a stable initial state becomes metastable when the field is changed. These are the units that contribute to the relaxation and transform isothermally and at constant field. The second aspect is to take into account the experimental procedure of field step-like variations: for both the increasing/decreasing series a low (0.005 T)/high (7 T) field has been applied between two successive step field values. The time elapsed between the intermediate low/high field value and the next constant step for the field is long enough to allow for the global change in magnetization that can take place due to the step field change (mostly in “instantaneous” or athermal mode); according to the observed magnetization changes, the isothermally transformed fraction at constant field is much lower than the athermal one during the step field change (see Fig. 2). Therefore, the relaxation rate or the number of relaxing units (Z / N{m}) appears to be proportional to the difference between the “new” transformed units upon the application of the field and those that have transformed under the previous stepwise field value. It is worth noting that Z is not proportional to the total change in magnetization from the high/low field to the selected constant field value. This difference essentially corresponds to the field rate of the transformation, dF/dH|T, in a continuous field increasing/
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decreasing constant temperature process inducing reverse/ direct MT. As commented above, Fig. 5 shows examples of continuous M(H) cycles at constant temperature. From these data, the derivative dM/dH can be obtained numerically. Fig. 6 presents the comparison between Z series at the different temperatures and the values of dM/dH. Therefore, the relation between Z and dF/dH|T is confirmed, as established in Eq. (4). In this sense, within this more “detailed” point of view, there is a clear parallelism in the relation between Z and the transformation rate in a continuous evolution of the field at constant temperature (Z(H) / dF/dH|T), and the previously found proportionality between Z (from relaxation in dwellings at constant temperature, constant field) and the transformation rate dF/dT|H in a continuous cooling/heating cycle. Therefore the relation between Z(T) and the rate dF/dT|H in temperature-induced transformations is found to be extended to a similar relation between Z(H) and dF/dH|T. This similarity anticipates a universal physical background in both temperature-driven and field-driven anisothermal transformations, and sets the close relationship between the isothermal transformation rate (Z / N{m}) when it is observable, and the total transformation rate dF/dT|H or dF/dH|T in the anisothermal transformations according to the definition given by Massalski [21]. Another point that deserves further discussion is linked to the presence of an extreme in the isothermal transformation rates (or a maximum considering the absolute values of Z in the decreasing field sequence). As has been mentioned above, the maximum is clearly seen for the three temperatures in the decreasing field sequence; nevertheless, it can be assumed that such a maximum also exists in the increasing field case: it is found at 200 K (see Fig. 4a), and the evolution of Z with the field up to 7 T at 150
(a)
and 120 K, anticipates that a maximum will be present at higher fields (in agreement with what has been discussed in the previous paragraph, dF/dH|T has to be bounded, as is the magnetization). These dependences of Z with the increasing/decreasing field sequences are perfectly coherent with the relation found between Z(H) and the ratio dF/ dH|T. The presence of a maximum in the values of Z means that there are limits for the field values, bounded at two sides, out of which the isothermal or relaxation behaviour does not take place, or is not observable at the level of the time scale allowed for the experimental conditions to reach a stepwise level of the field. In the present case, the isothermal isofield relaxation intervals (isothermal “windows”) for increasing/decreasing fields appear to have some degree of superposition in the range of studied temperatures. It is interesting to notice that the existence of isothermal windows (with logarithmic relaxation rates) has been found in a quite different system as Ce(Fe0.96Ru0.04)2 which also shows a metamagnetic transition (the low temperature phase has lower magnetization), as in the present alloy [22]. According to the shape of the M(H) curves of this alloy – taken at 5 K – in a continuous field change (see Figs. 3 and 6 of Ref. [22]), the isothermal behaviour window in increasing field sequences is found at higher fields and with a sharper cut-off limit than the corresponding window for the decreasing field sequence. These features are perfectly described taking into account the relation between Z(H) and dF/dH (Eq. (4)), which has been found in the present work for the Ni–Mn–In–Co alloy. Moreover, it is not surprising that this Ce-containing alloy presents disjoint relaxation windows for increasing/decreasing fields because of the much higher values of dM/dH (30 emu g1 T1), as compared to the present Ni–Mn– In–Co alloy, in most part of the M(H) curves. Taking into
(b)
(c)
(d)
Fig. 6. Comparison between Z and dM/dH obtained from continuous M(H) cycles as those shown in Fig. 5: (a) 200 K, H decreasing; (b) 200 K, H increasing; (c) 150 K, H decreasing; and (d) 150 K, H increasing.
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account the hysteresis in the M(H) cycle, this leads to a separate intervals of relaxation, again in agreement with Z(H) / dF/dH (Eq. (4)). It is necessary to mention here that for the same Ni– Mn–In–Co alloy, in temperature-induced transformations under constant field, relaxation dependences have been found, either in both or just in the cooling branch [9,10,16]. Therefore, it can be stated that the existence of certain (DT, DH) intervals of the driving variables for which the isothermal transformation occurs, is a general feature. Given a definite state of the system (including structure, degree of order, defects configuration, etc., which define the barriers to be overcome upon transformation, or in the thermodynamic sense, once the DG(T, H) landscape is determined), the relaxation can be observed or not, depending on the time dependence imposed to the variables T and H, and also on the available experimental time resolution for T and H. A very simple and typical example is given by the anisothermal transformations, which can be seen as athermal in continuous cooling/heating runs (for certain intervals of dT/dt under constant field, or dH/dt at constant T) or showing isothermal transformation steps at certain temperature (or field) dwellings within the continuous transformation range. It has to be taken into account that the transformation path F(T) or F(H) and consequently dF/dT or dF/dH can be modified depending on the dT/dt (dH/dt) rates, as the transformation includes time-dependent relaxations taking place during the continuous transformation (T induced or H induced). In the present Ni–Mn–In–Co alloy, under dT/ dt in the order of 10 K min1, or dH/dt 0.01 T s1 no significant changes in the transformation path are observed; the same occurs in the Ce-containing alloy [22]. This behaviour will be observed as far as the “athermal” component is contributing much more than the isothermal or relaxation one to the transformed fraction. Very recently, metastable states which relax towards the equilibrium have been found in an Ni–Co–Mn–Sn alloy [23]. According to this paper, these metastable states at low temperature are governed by the interplay of supercooled and kinetically arrested states, because none of them is the equilibrium state of the system. Therefore, is not surprising that under these conditions the system presents isothermal transformation as a mechanism to approach the equilibrium. The authors define the so-called kinetic arrest “bands”, given by some pairs of (H, T) values. The relations between these arrest bands and the isothermal or relaxation windows will be the object of future research. The fact that the number of relaxing units appears to be proportional to the global transformation rate, either dF/ dH|T or dF/dT|H, means that, upon a change in the driving field (T or H), there is no “a priori” separation between the parts of the sample (“units”) behaving in athermal (“instantaneous”) mode and those relaxing. Indeed, as they occur in the anisothermal transformations, relaxing units may become athermal or vice versa, by simply modifying
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the rate of variation in the driving fields, thus changing in a different way the free energy balance of the system. This feature is in perfect agreement with the common underlying physical mechanism of the phase transition in both types of units/transformation modes. At a macroscopic level, this means, for example, that the continuous transformation path is encountered once the isothermal or isofield dwellings are resumed, as has been observed in temperature-induced transformations in the same Ni– Mn–In–Co alloy considered here [10]. The relation between Z and the transformation rate allows for analysing some characteristics of the relaxation as a function of the progress of the transformation, i.e. as a function of the martensite fraction Xmart. This fraction, as mentioned above, can be easily calculated, for example at the beginning of the isofield dwellings, through the magnetization of the sample. In temperature-induced transformations under constant field, the construction of the socalled pseudo-transformation–temperature–time (TTT) diagrams, plotting martensite fraction as a function of 1/ |Z| has proved to be useful [16]. These diagrams are inspired in the conventional TTT diagrams used to characterize isothermal transformations, where every point (temperature–time pair) represents the condition to start the transformation and to obtain a given amount – usually a few per cent – of transformed material. In the pseudoTTT diagrams, the abscissa or time axis is replaced by the values of 1/|Z| as representative of the time rate at which the relaxation takes place. It has to be noticed that the pairs (temperature, 1/|Z|) represent the rate of accumulation of isothermally, isofield transformed phase for rather long time periods and at different degrees of the MT progress. In conventional TTT plots only the beginning of the MT and a small transformed fraction are considered. Pseudo-TTT diagrams have been obtained for the same Ni–Mn–In–Co alloy, for temperature-induced MT under constant fields [16]. It has been found that the pseudoTTT plot representing the martensite fraction as a function of 1/|Z| appears to be a common C-shaped curve, ranging from (practically) 0% to 100% transformed fraction, which includes different temperatures (Z values obtained at different martensite fractions in both direct and reverse MT), and applied magnetic fields. This case corresponds to a sample (water quenched, WQ, with a lower order degree than the SC samples used in the present work) that does not show arrest of the MT, at least up to the highest applied fields (4 T) in Ref. [16]. The nose of the common C-curve is situated very close to 50% of martensite fraction. On the other hand, in a continuous cycle, including complete direct/reverse MT, the maximum transformation rate corresponds to the maximum slope dM(T)/dT (temperature-induced MT at constant field, monitored by M(T)) or dR/dT (MT monitored by electrical resistance (R) vs. temperature measurements (T)); as this zone is close to 50% transformation, we found again the relation between Z and dF/dT, now in particular for |Z|max and dF/dT|max. This has been related to the amount of “active”
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austenite–martensite interfaces, which is largest close to 50% transformation in a complete cycle [16]. Fig. 7a compares the martensite fraction vs. 1/|Z| for a temperatureinduced MT under constant field (1 T) in a SC sample, with the SC sample subjected to the increasing/decreasing field procedure at 200 K. It is worth noting the similarity between these curves, even though the direct MT, in the decreasing field sequence, does not go from 0% to 100% martensite, as it starts at 80% austenite. It is obvious that the nose is close to 50% martensite. Looking at the increasing field points in Fig. 7a, they follow the same trend, taking into account that a field of 7 T is not enough to complete the reverse MT. However, the C-shaped curve for these points is shifted to higher values of 1/Z as compared with the decreasing field one, in correspondence with the values of Fig. 4a. From the results presented in Fig. 7a it could be concluded that for transformation paths, either T-induced or H-induced, which comprise a significant transformation amount (i.e. change in the martensite fraction of 0.8 or higher), the nose or maximum of |Z|
(a)
(b)
Fig. 7. Martensite fraction (Xmart) vs. 1/|Z| for temperature-induced transformation under constant field (1 T), and for increasing/decreasing fields at (a) 200 K and (b) 150 K.
is close to a 50% martensite. As has been stated in Ref. [16] for T-induced MT under constant field, this means that the maximum number of transforming units is related to the largest area/number of moving martensite–austenite interfaces, which corresponds to 50% transformation in a complete (or close to complete) transformation cycle. This is one of the significant points of the present work, as these results generalize to H-induced transformations those of Ref. [16] obtained for temperature-induced MT at constant fields. This fact means that a common physical mechanism is responsible for the time-dependent transformations. Let us discuss the relation between the nose position (|Z|max) in pseudo-TTT diagrams and the fractions of martensite in the cases were the amount of transformation is reduced. Fig. 7b shows again Xmart vs. 1/|Z| for the increasing and decreasing field sequences at 150 K, together with the data of MT under constant field (1 T) in a SC sample, already shown in Fig. 7a, for comparison. First, it can be noticed that the increasing field series at 150 K follows the same C-shaped curve as for 200 K, although the minimum of 1/|Z| does not correspond to the nose. This is in agreement with the fact that a field of 7 T is far from being able to complete the reverse transformation at 150 K, and the maximum of Z is not reached at this temperature (as seen in Fig. 4b). The same trends are observed at 120 K. For the decreasing field sequence a kind of nose appears. Even though the total fraction of transformed martensite is not far from 0.3, a maximum of |Z| appears in correspondence with the shape of the decreasing field curve, as seen in Fig. 5 (the shape of the increasing field curve obviously explains that up to 7 T a maximum slope is not reached, and therefore the absence of a nose), in agreement with Eq. (4). Keeping once more in mind the relation between Z and N, it can be concluded that, even in the cases where the total transformed fraction is low, the value of Z (/ N / dF/dH) is related to the number of “active” martensite–austenite interfaces. 5. Conclusions Relaxation towards the equilibrium state occurs upon isothermal magnetic field step variations, both in increasing and decreasing field sequences. In all cases the time dependence, followed through the magnetization M(t), appears to be logarithmic. Therefore, the analogy with the magnetic after-effect allows analysing the slope of logarithmic dependence, Z (or viscosity coefficient), in terms of the number of relaxing units and temperature (Eq. (3)). It has been found that the rate of relaxation or the number of relaxing units (Z / N{m}) appears to be proportional to the transformation rate upon isothermal, continuous field change Z(H) / dF/dH|T / dM/dH|T. Therefore the previously established proportionality between Z(T) (relaxation in dwellings at constant temperature within a temperature cycle), and the transformation rate dF/dT|H in a continuous cooling/heating cycle, is
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found to be generalized to a similar relation between Z(H) and dF/dH|T. This fact points out a common physical origin of relaxations in both temperature- and field-induced MT. Following the analogy with the magnetic after-effect, the relaxations take place through a variety of energy barriers, leading to logarithmic time dependence. The presence of a maximum in Z sequences implies that there are limits for the field values, bounded at two sides, out of which the isothermal/isofield or relaxation behaviour does not take place, or is not observable at the level of the time scale allowed for the experimental requirements to reach a stepwise level of the field. In the present case, the isothermal isofield relaxation intervals (isothermal “windows”) for increasing/decreasing fields appear to have some degree of superposition in the range of the studied temperatures. The existence of isothermal windows has been found in different systems as Ce(Fe0.96Ru0.04)2. The relaxation features of this system are perfectly explained taking into account the relation between Z(H) and dF/ dH, which has been found for the present Ni–Mn–In–Co alloy. The relation between Z and the transformation rate allows for analysing some characteristics of the relaxation as a function of the progress of the transformation, i.e. as a function of the martensite fraction. As in temperature-induced transformations under constant field, the socalled pseudo-TTT diagrams can be constructed in the case of isothermal/isofield relaxations. The relation between the maximum relaxation rate (minimum of 1/|Z| or nose in the TTT diagram) and the number of interfaces appears to be also valid for the present case, thus extending and generalizing to magnetic-field-induced MT the results found in temperature-induced transformations under constant field. In summary, relaxations towards the equilibrium have been found to have a common physical background in both temperature-driven (Z(T) / dF/dT|H) and field-driven (Z(H) / dF/dH|T) transformations. The existence of relaxation windows has been understood directly from the experimental results, and the shape of pseudo-TTT diagrams has been related to the number of “active” units participating in the relaxation. Acknowledgements Partial financial support from MINECO and FEDER (Projects MAT2011-28217-C02-01 and MAT2012-37923-
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